Paradise Trinity Day

[Amigoo]

Re: VoSoTaS design
About the length, (2(sqrt(Pi))/Pi)(sqrt(2)) ...

[ > 1.4142135623730950488016887242097.. ] sqrt(2)
[ x 1.1283791670955125738961589031215.. ] 2(sqrt(Pi))/Pi
[ = 1.595769121605730711759784239736... ] (2(sqrt(Pi))/Pi)(sqrt(2)) *
[ / 1.2533141373155002512078826424055.. ] sqrt(Pi/2)
[ > 2.0 ]

* (2(sqrt(Pi))/Pi)(sqrt(2)) of the blue CSC set defines a square
whose inscribed circle is the circle of the square where s = sqrt(2)
and inscribing circle is the circle of the square where s = 2.0.

CSC = Circle inscribed in Square inscribed in Circle
Two CSC sets in VoSoTaS: one blue, one magenta.

Rod ... ...

[Amigoo]

Re: VoSoTaS design
More about the length, (2(sqrt(Pi))/Pi)(sqrt(2)) ...

[ > 1.4142135623730950488016887242097.. ] sqrt(2)
[ x 1.1283791670955125738961589031215.. ] 2(sqrt(Pi))/Pi
[ = 1.595769121605730711759784239736... ] (2(sqrt(Pi))/Pi)(sqrt(2)) *
[ / 1.2533141373155002512078826424055.. ] sqrt(Pi/2)
[ > 2.0 ]

* identifies circles of squares where s = sqrt(2), 2.0
and side of inscribed square for D = 2(2(sqrt(Pi))/Pi)

Say what?

For D = 2(2(sqrt(Pi))/Pi),
length of side of inscribed square
= 2(2(sqrt(Pi))/Pi) / sqrt(2)

2.256758334191025147792317806242..
/ 1.4142135623730950488016887242097..
= 1.595769121605730711759784239736..

Rod

[Amigoo]

Re: VoSoTaS design

For D = 2(2(sqrt(Pi))/Pi),
length of side of inscribed square
= 2(2(sqrt(Pi))/Pi) / sqrt(2)

Exploration of the inscribed square(s) in a circle of this diameter
inspired more creativity. Now, when the design is flipped vertically,
an abstract figure of a divine being seems to appear ... and this
complements the acronym, VoSoTaS.

If Gaia is the personification of the Earth, this divine being
might be the personification of our local universe.

Rod ... ...

[Amigoo]

Re: Pi of Pi design

"Oh where, Oh where is 2(sqrt(Pi))/Pi?" I wondered ...
as would all believing squared circle geometers.

The geometry of VoSoTaS offered the ideal Cartesian
neighborhood for a wandering search, then discovery!

2(sqrt(Pi))/Pi (= 1.1283791670955125738961589031215..),
is the lower red diagonal having the same length as the
radius of the largest circle where D = 2(2(sqrt(Pi))/Pi)
(= 2.256758334191025147792317806242..).

Rod

[Amigoo]

Re: Pi of Pi design

"Oh where, Oh where is 2(sqrt(Pi))/Pi?"

The vertical yellow line?
Length = 2 [side of square of D = 2(2(sqrt(Pi))/Pi)]
Savor the Pi of Pi ... if you believe in squared circles.

Rod ... ...

[Amigoo]

Re: Pi of Pi design (renamed to "i of Pi")
(see in: http://aitnaru.org/images/Tripartite_Soul.pdf )

"i" alludes to both geometric embellishment (looks like a colorful eye) and to
2(sqrt(Pi))/Pi [equals 2(sqrt(1/Pi))], a value common to squared circle geometry.

This geometry is visually challenging:
Imagine that the yellow scalene triangle (with inscribed red isosceles triangle) moves as a pendulum with the pivot point at the top and two longest sides changing in length. The lower red triangle moves with its hypotenuse continually parallel to the left side of the larger red triangle. The circle(s) is squared at a precise position of the scalene triangle, but what is this "impossible" geometric magic? ... perhaps revealed only through the "i of Pi" lens.

Rod

[Amigoo]

Re: "i of Pi" design

More reflection on the "i of Pi" ...
aka 2(sqrt(Pi))/Pi and 2(sqrt(1/Pi))
= 1.1283791670955125738961589031215..

Obviously, diameter / 2(sqrt(1/Pi)) = side of circle’s square
... and this reveals an intriguing geometric relationship:

Squared circle relationship of diameters 2.0, 5.0

2.0
/ 1.1283791670955125738961589031215.. 2(sqrt(1/Pi)
= 1.772453850905516027298167483342.. sqrt(Pi)
5.0
/ 1.1283791670955125738961589031215.. 2(sqrt(1/Pi)
= 0.44311346272637900682454187083529.. sqrt(Pi)/4

Rod .. ...

[Amigoo]

Re: "i of Pi" design

The correct
Squared circle relationship of diameters 2.0, 5.0

2.0
/ 1.1283791670955125738961589031215.. 2(sqrt(1/Pi)
= 1.772453850905516027298167483342.. sqrt(Pi)
5.0
/ 1.1283791670955125738961589031215.. 2(sqrt(1/Pi)
= 4.4311346272637900682454187083529.. 10(sqrt(Pi)/4)

Rod

[Amigoo]

Re: "i of Pi" design

To restate (and acknowledge "square root")

"Simple Simon met a pie man, parting from the fair.
Said the pie man to Simple Simon: 'My pies are in repair
to lockers cold, thus wrapped - not sold; leftover fare is fair
for sweet repose, then new expose of pie beyond compare!'"

On post Leftover Pi Day, 2015 ...
Whether known as the "i of Pi" or "Pi beyond compare",
2(sqrt(1/Pi)) is another contender for tasty Pi since
Diameter / 2(sqrt(1/Pi)) = Side of circle's square
(aka the "square root of the circle's square")

2(sqrt(1/Pi)) = 1.1283791670955125738961589031215..

Rod

[Amigoo]

Re: "Acos In sight" design

“To sleep, perchance to dream”,
then awaken … and believe.

Exploration of the "i of Pi" geometry.

Rod ... ...

[Amigoo]

Re: "Acos In sight" design
“To sleep, perchance to dream”,
then awaken … and believe.

Regarding the two right triangles ...

1.0 length of long side
/ 1.1283791670955125738961589031215.. 2(sqrt(1/Pi)), length of hypotenuse
= 0.88622692545275801364908374167057.. sqrt(pi)/2

acos(0.88622692545275801364908374167057..)
= 27.597112635690604451732204752339.. degrees, cosine angle

Rod

[Amigoo]

Re: http://www.answers.com/Q/What_is_Leftover_Pi_Day (updated)

Entrepreneur alert: If the math is correct,
this might become a popular T-shirt:

Diameter / 2(sqrt(1/Pi))
= Pi are square!

(display the poem on the other side ...
but check for any copyright claims)

Rod ... ...

[Amigoo]

Re: "Acos In sight" design
“To sleep, perchance to dream”,
then awaken … and believe.

This quote can be appreciated on two levels:
- believe that squared circles exist in our material universe.
- experience the mortal transition to the mansion worlds
... then believe that life continues after death.

Rod

[Amigoo]

Re: Ratios of Pi design (in Tripartite_Soul)

"Convincing geometry of squared circles,
highlighting ratios of 2(sqrt(1/Pi)) and sqrt(Pi)."
(red geometric object contrasts two sets of line lengths)

14:14 has interesting balance (14 pages in each file):
http://aitnaru.org/images/Tripartite_Soul.pdf
http://aitnaru.org/images/Veritas_27597e.pdf

Rod ... ...

[Amigoo]

Re: Ratios of Pi design

Three values are now contrasted in
the red Pythagorean right triangle (top of design):
2(sqrt(1/Pi)), 1.1283791670955125738961589031215..
sqrt(Pi), 1.7724538509055160272981674833411..
2.0

Rod

[Amigoo]

Re: Ratios of Pi design

About the light blue "looking glass" ...
Length of circle's diameter (along "handle") is from ratio
of yellow vertical line and lower yellow diagonal:

1.7724538509055160272981674833411.. sqrt(Pi), vertical line
1.4142135623730950488016887242097.. sqrt(2), diagonal line

sqrt(Pi) x sqrt(2)
= 2.506628274631000502415765284811.. 2(sqrt(Pi) / sqrt(2))
sqrt(Pi) / sqrt(2)
= 1.2533141373155002512078826424055.. side of an isosceles right triangle

2.506628274631000502415765284811..
- 1.2533141373155002512078826424055.. = 2.0, length of circle's diameter

Rod

[Amigoo]

2.506628274631000502415765284811..
- 1.2533141373155002512078826424055.. = 2.0

I had to turn my head so much to see "/" instead of "-"
that the point of these numbers was lost in transit
(if there was a point).


Here's the complete bedtime tale
with similar points (if there is a point):

1.4142135623730950488016887242097.. sqrt(2)
x 1.4142135623730950488016887242097..
= 2.0

1.4142135623730950488016887242097..
/ 1.4142135623730950488016887242097..
= 1.0

2.0 / 1.0 = 2

1.7724538509055160272981674833411.. sqrt(Pi)
x 1.4142135623730950488016887242097.. sqrt(2)
= 2.506628274631000502415765284811..

1.7724538509055160272981674833411..
/ 1.4142135623730950488016887242097..
= 1.2533141373155002512078826424055..

2.506628274631000502415765284811..
/ 1.2533141373155002512078826424055..
= 2.0

2 x 1.4142135623730950488016887242097..
= 2.8284271247461900976033774484194..

2 / 1.4142135623730950488016887242097..
= 1.4142135623730950488016887242097..

2.8284271247461900976033774484194..
/ 1.4142135623730950488016887242097..
= 2.0


Oh! Now, I remember a point:
sqrt(Pi) exists between sqrt(2) and 2.0
Hmmm ... "= 2.0" has significance!

Rod ... ...

[Amigoo]

Re: Scalene Of Course design - “Full speed ahead!”
(re: http://aitnaru.org/images/Tripartite_Soul.pdf )

"Scalene Of Course" removes any doubt about which geometric object
in a Cartesian neighborhood is most definitive of squared circles!

A scalene what? The triangular object with Pythagorean siblings.

Rod

[Amigoo]

Re: Scalene Of Course design - “Full speed ahead!”

"Scalene Of Course" removes any doubt about which geometric object
in a Cartesian neighborhood is most definitive of squared circles!

A scalene what? Two overlapping scalene triangles
that form an isosceles trapezoid. Who knew?!

Now, "Of Course" refers to the midpoint-to-midpoint lines association
of the larger trapezoid with the angle-to-angle lines of the smaller
... of course.

Rod ... ...

[Amigoo]

Re: Scalene Of Course design ("within Trapezoid")

Pop Quiz:

If the diameter of the largest circle equals 4.0
and a circle were drawn around the inner triangle
(scalene, inscribed in circle) having two blue
sides and one red ...

What is the diameter of the circle drawn?

Hint: one side of scalene has length = sqrt(2),
another side has length = sqrt(Pi).

Rod

[Amigoo]

Re: Triunity of Trapezoids design
(aka Scalene Of Course "within Trapezoid")

What can be the purpose of such geometric complexity?
Evidence to "prove" that the circle cannot be squared?

This post Lindemann millenium should be equally instructive!

Rod ... ...

[Amigoo]

Re: Triunity of Trapezoids design
(aka Scalene Of Course "within Trapezoid")

"New and Improved" (redrawn to verify geometry)

Required Disclaimer:

Caution: This geometry is more difficult than it looks.
Construct only in flawless Cartesian neighborhoods.
Any opinions expressed are not those of this network.

Rod

[Amigoo]

Re: Trinity of Trapezoids design
(aka “circling the square”)

Hmmm ... which is "impossible"?
Squaring the Circle or Circling the Square?

“To sleep, perchance to dream”,
then awaken … and believe.

Rod

[Amigoo]

Re: Squared Circles 101
(aka "child's play")

Final 2015 exploration of the defining geometric value
2(sqrt(1/Pi)) = 1.1283791670955125738961589031215..

Rod ... ...

[Amigoo]

Re: Squared Circles 101
"2(sqrt(1/Pi)) in vitro"

Since 2.0 (a diameter)
/ 1.7724538509055160272981674833411.. sqrt(Pi)
= 1.1283791670955125738961589031215.. 2(sqrt(1/Pi))

Then, a circle of diameter 2(sqrt(1/Pi)), is squared
by a right triangle having a long side = 1.0
... apparently.

Rod